350 requires a type rating correct?
In which case you can't (legaly) log anything unless you have at least a SIC type. Will you be paid? The insurance company requires so are they looking for a bright eyed bushy tailed free pilot? What SIC training does the insurance company require?
Did they mention anything about sending you to training?
Be careful that you don't end up being the free luggage loader and airplane washer and refueler while they do 'you a favor'.
Something doesn't fit here.
The Captain is SE qualified in the airplane but the insurance company requires a SIC. Anything can be insured, even a private pilot in a 707 so it's a matter of how much premium they're willing to pay.

But I digress.
To answer your question.
Keep a written record as far as dates, tail number and if you had any stick turn or not. It's still experience even though it's not 'loggable'.
You'll impress any recruiter by knowing when you can or cannot log the time.
You could even buy a separate (cheap) PPL style logbook and have the Captain sign the flights.
Double bookkeeping if you will but don't use the hours for a certificate or rating. Just extra experience.
Clear as mud?